Outer Limits of Reason (42 page)

Relativity theory comes in two flavors. In 1905, Einstein formulated the
special theory of relativity
, which deals with the universe without gravity or acceleration. Later, in 1914, he generalized this work to the
general theory of relativity
, which deals with gravity and acceleration. I start with special relativity and work my way toward general relativity.

The central idea of relativity theory is that properties of the physical universe depend on how they are measured. There are no absolute measurements. This is in sharp contrast to our naive notions about the universe. We usually say a person that we are looking at has an exact height. They will be perceived differently when we are at different distances. When we are far away they look small and when we are near they look large. In this simple case, we can say that there is an absolute height but there are different relative heights. In contrast to the usual notions, relativity theory tells us that properties of an object really are different depending on how they are viewed. There are no absolutes.

Let us begin by asking a simple question: What is the length of the coast of Norway? It is not hard to get an answer. Simply take a map and a ruler and measure it. The problem is that the coast is not a straight line. There are many curves and turns that make it hard to get an exact number. If you get a larger map or you measure it with a smaller ruler and are able to take into account more of the nooks and crannies of the coast, you will find that the length of the coast will be longer. One could take this question very seriously and actually walk along the coast of Norway and measure it while walking around all the magnificent fjords. Measuring the coastline like this would further increase the length of the shoreline. If one were to ask an ant to walk the coastline of Norway, the coast would be even longer.
³⁵
What is the real length of the coast of Norway? The answer is that the length depends on how it is measured. This strangeness is called the
coastline paradox
, but of course the idea has nothing to do with Norway or a coastline. We could have asked that question about many physical objects.

It is important to realize that this thought experiment is arguing against absolute lengths. One might believe that there is really an exact length of the coast of Norway and that by using better and better instruments we will get better and better approximations to this exact length. This is false. Rather, the length depends on how it is measured. We are going to find similar phenomena within relativity theory.

Galilean Relativity

Einstein's relativity theory was based on the work of earlier giants. Galileo described how our perceptions of the laws of physics remain unchanged when there is constant movement. He discussed movement in an important transportation vehicle for his generation: boats. Different experiments were performed on both a stationary and a moving boat. His writing is so beautiful and clear that it is worth quoting at length:

Shut yourself up with some friend in the main cabin below decks on some large ship, and have with you there some flies, butterflies, and other small flying animals. Have a large bowl of water with some fish in it; hang up a bottle that empties drop by drop into a wide vessel beneath it. With the ship standing still, observe carefully how the little animals fly with equal speed to all sides of the cabin. The fish swim indifferently in all directions; the drops fall into the vessel beneath; and, in throwing something to your friend, you need to throw it no more strongly in one direction than another, the distances being equal; jumping with your feet together, you pass equal spaces in every direction. When you have observed all of these things carefully (though there is no doubt that when the ship is standing still everything must happen this way), have the ship proceed with any speed you like, so long as the motion is uniform and not fluctuating this way and that. You will discover not the least change in all the effects named, nor could you tell from any of them whether the ship was moving or standing still. In jumping, you will pass on the floor the same spaces as before, nor will you make larger jumps toward the stern than towards the prow even though the ship is moving quite rapidly, despite the fact that during the time that you are in the air the floor under you will be going in a direction opposite to your jump. In throwing something to your companion, you will need no more force to get it to him whether he is in the direction of the bow or the stern, with yourself situated opposite. The droplets will fall as before into the vessel beneath without dropping towards the stern, although while the drops are in the air the ship runs many spans. The fish in the water will swim towards the front of their bowl with no more effort than toward the back, and will go with equal ease to bait placed anywhere around the edges of the bowl. Finally the butterflies and flies will continue their flights indifferently toward every side, nor will it ever happen that they are concentrated toward the stern, as if tired out from keeping up with the course of the ship, from which they will have been separated during long intervals by keeping themselves in the air.
³⁶

Galileo was describing many different experiments demonstrating that the laws of physics cannot tell the difference between a boat standing still and a moving (unaccelerating) boat.
³⁷

Einstein discussed many similar experiments on trains, a major transportation vehicle for his generation. However, I will talk about cars since they are a little more contemporary. While you are a passenger in a car going 50 miles per hour, throw a small ball upward. If the car is not accelerating, decelerating, or making a sharp turn, the ball will gently fall right back into your hands. This is actually an amazing fact since your hand has moved forward a couple of yards in the few seconds that the ball was in the air. In fact, the ball would act the same way if the car was not moving at all. This is exactly Galileo's point about the ship: one cannot detect movement by looking at how things move inside the boat/train/car. (You can, of course, look out the window to determine if you are moving.)

Now think of people standing on the curb watching your car and your ball. What do they see? They do not simply see the ball rising and falling. Rather, they see the ball rising and going forward. After all, the car is moving at 50 miles per hour and the ball will go forward with it. The ball will land in your hand in a few seconds. The point is that as a passenger, you can do some calculations, see that you are throwing the ball straight up, and calculate when it will land. At the same time, the people on the curb see the ball go up and forward, perform some calculations, and determine when and where the ball will land. Although the calculations are different, the laws of motion are the same.

Following the ideas of Galileo, Einstein assumed these results were true for any observer looking at the physical universe:

Postulate 1:
  All observers at a constant speed must observe the same laws of motion.

Since this idea was already known to Galileo, this postulate is known as
Galilean relativity
. However, Einstein went further with these ideas.

Special Relativity

To understand Einstein's theory of special relativity we must start with a discussion of the speed of light. The very fact that light travels at a finite speed and is not instantaneous is counterintuitive. Simply turn on the light and it seems that the room instantly lights up. Nevertheless, scientists in the seventeenth century realized that light is not instantaneous but travels at a finite speed.

The first experiment that attempted to calculate the speed of light was done in 1676 by the Danish astronomer Ole Rømer (1644–1710). The idea is pure genius and worthy of our attention. Rømer was using this newfangled invention called a telescope to observe the planet Jupiter and one of its moons called Io. The moon rotates around Jupiter at a fixed speed. That means that the time that Io goes behind Jupiter (eclipsed) and the time that it emerges from behind Jupiter (moonrise) should be set times. However, the astronomer noticed that when the Earth was far away from Jupiter, Io's appearance was delayed (see
figure 7.22
).

He reasoned that the cause for the delay was that it took longer for the light reflected off Io to reach the Earth. From his knowledge of the distance of the Earth from Jupiter and his knowledge of the size of the orbit of the Earth, Rømer was able to calculate the speed of light. While his calculations were off, the idea was brilliant and led other scientists to perform more exact experiments.

Eventually it was determined that light (in a vacuum) travels at about 186,000 miles a second. There was, however, something shocking about the speed of light, namely, that this speed is constant regardless of the speed of the source of the light or the speed of the observer. This is contrary to any other phenomenon in the universe. If you are traveling in a car at 50 miles per hour and another car is traveling in the same direction at 30 miles per hour, you will perceive the other car as going only 20 miles per hour. If you are traveling in a car at 50 miles per hour and a car is coming toward you at 30 miles per hour, you will perceive it as coming toward you at 80 miles per hour. This is true for cars and other objects in the universe. It is
not
true for light. If you are traveling, regardless of whether the source of the light is coming toward you or going away from you, it will appear to be going at 186,000 miles per second.

Einstein realized the consistency of the speed of light by looking at the equations that describe light (Maxwell's equations for electromagnetic waves) and seeing that the speed of the observer and the speed of the source of the light are “not even in the equation(s).”

There were also experimental results that showed that the speed of light will always be perceived at the same rate. The simplest experiment was described in 1913 by the Dutch astronomer Willem de Sitter (1872–1934). He considered a binary star system—that is, two stars close enough that their forces of gravity pull each other into a spin, as in
figure 7.23
.

If light did not travel at a constant speed, it would move faster from an approaching star and more slowly from a retreating star. Such comings and goings happen all the time with a binary star system. If light did not travel at a constant speed, then the light coming from the star approaching the Earth would reach the Earth before the light coming from the retreating star. In that case, the light would come in a scrambled form. De Sitter reported that no such scrambling occurs. He concluded that the speed of light is constant regardless of the velocity of the source of the light.

As an interesting sidebar, the consistency of light is used in making certain definitions of distances and times. The units of measurements that we use, such as mile, foot, inch, meter, hour, minute, or second, are derived from cultural and historical factors. Researchers would like a more scientific way of describing these units. Since the speed of light is consistent, it is used to determine the official scientific definition of certain lengths. Given that light travels at precisely 299,792,458 meters per second, we can take it as a definition that 1 meter is the distance that light travels in 1/299,792,458 second. What is a second? To answer this, researchers have considered certain vibrations that are done by a cesium-133 atom. A second is defined as the time it takes this atom to make 9,192,631,770 vibrations. This number was chosen because it matches up with what we historically know is a second. These two units of measurement give us exact technical definitions of length and time. But we will see that these definitions are slightly misleading.

Einstein took the consistency of the speed of light as a postulate about the universe:

Postulate 2:
  All observers will always view the speed of light at the same rate.